- #1

- 259

- 45

[tex]\frac{1}{2}\frac{\partial }{\partial \left ( \partial _{\mu}\varphi \right )}\left ( \partial _{\mu}\varphi \right )^2[/tex]

wherein ##\varphi(x,y,z,t)## is a scalar field. My approach to this was as simple as it was naive :

[tex]\frac{1}{2}\frac{\partial }{\partial \left ( \partial _{\mu}\varphi \right )}\left ( \partial _{\mu}\varphi \right )^2=\frac{1}{2}\frac{\partial }{\partial \left ( \partial _{\mu}\varphi \right )}\left ( \partial _{\mu}\varphi \right )\left ( \partial _{\mu}\varphi \right )[/tex]

which evaluates to ##\partial _{\mu}\varphi## via the product rule. However, this is where I get stuck, because the answer is wrong - the correct approach should have been

[tex]\frac{1}{2}\frac{\partial }{\partial \left ( \partial _{\mu}\varphi \right )}\left ( \partial _{\mu}\varphi \right )\left ( \partial ^{\mu}\varphi \right )[/tex]

which apparently evaluates to ##\partial ^{\mu}\varphi## ( though I have difficulties with that as well, but that's a separate issue ), and leads to the correct equations of motion. My question is : why is ##\left ( \partial _{\mu}\varphi \right )^2=\left ( \partial _{\mu}\varphi \right )\left ( \partial ^{\mu}\varphi \right )## and not ##\left ( \partial _{\mu}\varphi \right )^2=\left ( \partial _{\mu}\varphi \right )\left ( \partial _{\mu}\varphi \right )## ? I know that this is probably something very elementary, so please don't laugh at me, but I genuinely don't get it.